March 10, 2008

Case Western Reserve mathematician tells how to pair Bayesian statistics with scientific computing

For the majority of people, the gates of scientific computing open to numbers only, and anything not expressed in quantitative terms is of no help.

This, Case Western Reserve University mathematician Daniela Calvetti says, seems like a waste of good information by not taking advantage of people's hunches and beliefs about problem solving and what the results should be. Combining a hunch with numbers could help computers do a better job in solving problems.

The researchers explore the synergy between the disciplines of
Bayesian statistics and numerical analysis. The Bayesian statistics provides the framework for taking advantage of the "hunch" part of the process, while the numerical analysis quantifies it in a manner usable in the computations.

For example, the computer routinely transforms images into numbers, computes the numbers and then returns answers in the form of images.

"If we could just pass on to the computer algorithms some of the knowledge that people have or acquire about the problems they are solving, we would come a long way," Calvetti stated.

For example, if a doctor looking at a patient's mammogram might not know what to expect, but his partial knowledge will come into action if by accident the mammogram is replaced with an x-ray of a broken knee.

Through discussions with colleagues around the world and with her
students in her classes, Calvetti said she came to realize that
"researchers have some vague idea of what was going on even before measuring anything, and they have data to process, but often do not know how to put the two steps together."

In a step-by-step way, Calvetti shows them how to do this and
includes practice software applications to let the reader experiment with the process and adapt the algorithms to individual research problems.

Calvetti added that she made special efforts to find a language which conveyed the mathematical ideas without following the traditional theorem and proof format.

The book will become the textbook for a fall 2008 course that
Calvetti will teach on the integration of numerical and statistical methods for math, statistics, biomedical engineering and computer science graduate students. Students from other disciplines may also find applications in the course of interest.

Calvetti, a specialist in scientific computing and inverse problems, set out to streamline algorithms to make them perform faster and better. Her goal is to make it possible to run heavy duty computational tests on a personal computer, because not everyone has super-computing capabilities.

During that time, Calvetti experienced the limitations of traditional computational methods, because of the scarcity and variability of the data.

What clued a solution for her was during meetings where computer simulation results were discussed. "Doctors and biologists could just look at numbers and know right away they were incorrect. They had the extra bit of information that could help the computations but they were not in a format that could be immediately used, she said.

"The idea was to take advantage of that knowledge and import it into the numerical computations," she explained.

While refining the idea over a couple of years through writing as many as 20 papers, making conference presentations and teaching a preliminary version of the course that she will offer this coming fall, Calvetti said the process began to emerge and bring the two disciplines together.

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